The crew consists of: Captain Dan Holland, First Officer Lieutenant Charlie Pizer, journalist Harry Booth, ESP-sensitive scientist Dr. Palomino, is returning from a deep space exploration mission. There are multiple forms that the metric can take.An exploratory spaceship, the U.S.S. So, now that we’ve made it through this, let’s move on to the Schwarzschild metric. However, it’s all limited by just how much the spacetime is curved. Now, we, light, planets, the stars, and virtually everything in the cosmos takes the shortest path between two different events in spacetime. We can also if we’d like, calculate how much our spacetime has curved - at least in the region of space around bodies with a mass. We know that gravity causes a curvature in spacetime. Well, mathematically, it can certainly be done. Could we, without having to escape to a “fifth” dimension, realize whether we’re in flat spacetime or curved spacetime. We can apply that question to ourselves too. The interesting question though is that could an object stuck on the surface of a sphere tell that it’s on a curved surface without having to look at it in three dimensions. That will simply be a straight line - meaningless if we’re stuck on a sphere. This, evidently, is not the same as dx² + dy² + dz² = ds². In this situation, if we’re confined to the surface of the sphere - that is, we can’t drill a hole inward or fly up into the air - the only valid distance we can measure is an arc length. Let’s say that the two points are confined to the surface of a sphere and for that matter, so are we. This time, however, we’ll work with a limitation. So, we’ll move back to 3D space and imagine that we’re trying to measure the distance between two points on a 3D grid. This is usually difficult to imagine in four dimensions - partly because, well, we’re terrible at visualizing four dimensions. The shortest distance between two points in curved space is actually what we’d call a geodesic. We’ve made an unsaid assumption here: the shortest distance between two points in space is a straight line. However, the issue with this is that our equation assumes that space-time is flat. What’s important now is we’ve calculated a framework for finding the interval between two events in four-dimensional space-time. We haven’t even got to the black hole equation yet. It’s that we must account for an extra term when we consider higher dimensions. The idea remains the same regardless of its inclusion. Not really important but that’s what we’ll call it nonetheless.Īnyway, we won’t be delving into the consequences of having a -c² term here. This is just embedded into the fundamental idea of spacetime. As long as we’re consistent, we’ll be met with the same answer.Īnother caveat we want to consider when we invoke spacetime is that we’re not finding the interval between two points anymore we’re finding the interval between two events. It doesn’t really matter if we choose negative for the time and positive for the space or positive for the time and negative for the space. The negative sign with the c² is irrelevant for now. Which, when we apply to Pythagoras’ theorem, we need to square. You then only move in the time-like direction, and in this direction, you move with the speed of light. Relative to yourself, you do not move through space, so these velocities are zero. So, our dt is scaled up by the speed of light. This is because we move through time at the speed of light. Now, when we modify our formula and account for the time coordinate we include a c² term. In this case, however, we’re considering an interval. By the way, the reason why I called it the interval rather than the distance is because “distance” implies a spatial perspective. This is exactly how we accounted for the third dimension when we went from the 2D grid to the 3D one. In other words, if we wanted to find the interval between two points in four-dimensional space, we’d need to account for the fourth dimension.
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